option mispricing around nontrading...

THE JOURNAL OF FINANCE • VOL. LXXIII, NO. 2 • APRIL 2018

Option Mispricing around Nontrading Periods

CHRISTOPHER S. JONES and JOSHUA SHEMESH∗

ABSTRACT

We find that option returns are significantly lower over nontrading periods, the vastmajority of which are weekends. Our evidence suggests that nontrading returns can-not be explained by risk, but rather are the result of widespread and highly persistentoption mispricing driven by the incorrect treatment of stock return variance duringperiods of market closure. The size of the effect implies that the broad spectrum offinance research involving option prices should account for nontrading effects. Ourstudy further suggests how alternative industry practices could improve the efficiencyof option markets in a meaningful way.

A VARIETY OF INSTITUTIONAL and psychological factors suggest that portfolio riskand return may differ between periods of trading and nontrading. The stockmarket is more volatile over trading periods than over nontrading periods,possibly due to a lower rate of private information revelation (French and Roll(1986)). It is also profoundly less liquid outside of regular trading hours, whichshould drive a wedge between average returns over trading and nontradingperiods (French (1980), Longstaff (1995), Kelly and Clark (2011), Cliff, Cooper,and Gulen (2008)). Prolonged periods of nontrading may also give investorswith limited attention a chance to process stale information, whether containedin firm earnings announcements (Dellavigna and Pollet (2009)) or news moregenerally (Garcia (2013)). Finally, traders may have a strong desire to exit themarket over nontrading periods, with no open positions, particularly for morespeculative strategies or for longer periods of nontrading (Hendershott andSeasholes (2006)).

∗Christopher S. Jones is with the Marshall School of Business at the University of SouthernCalifornia. Joshua Shemesh is with the Department of Banking and Finance at the MonashBusiness School. We are grateful for comments from Editors Cam Harvey and Ken Singleton,two anonymous referees, Mike Chernov, Dan Galai, Bruce Grundy, Larry Harris, Robert Hudson,Blair Hull, Nikunj Kapadia, Spencer Martin, Dmitriy Muravyev, Yigal Newman, Oguz Ozbas, NickPolson, Allen Poteshman, Eric Renault, David Solomon, Selale Tuzel, Mark Weinstein, FernandoZapatero, and seminar participants at USC, the University of Queensland, Hebrew University, the2009 Montreal Financial Econometrics Conference, the 2009 Financial Management Associationmeetings, and the 2010 American Finance Association meetings. Most of the research was donewhile Shemesh was affiliated with the University of Melbourne. Shemesh thanks the Sanger Chairof Banking and Risk Management for financial support. Both authors state that their work on thisresearch was not supported financially by any nonacademic sources and that no organization orindividual has had any right to review this work.

DOI: 10.1111/jofi.12603

861

862 The Journal of Finance R©

While the above effects have been documented to varying degrees in themarket for individual stocks or stock indexes, one might expect most if notall to be more pronounced in the market for options. Predictable differencesin volatility should have first-order effects on option prices (see also Dubinskyand Johannes (2006)), while varying liquidity should affect not only the optionend user but also the market maker who must delta-hedge his underlyingexposure in the stock market (e.g., Cetin et al. (2006)). Limited attention maybe more problematic in option markets, where small changes in fundamentalscan have large valuation effects. Finally, option trades can be extremely risky,particularly for net written option positions, pushing some traders to withdrawcompletely during overnight or weekend periods.

Among these effects, it is the desire to close positions over the weekend thatChen and Singal (2003) conclude was responsible for the weekend effect instocks. The weekend effect, or the tendency of stock returns to be low over theweekend, is perhaps the most widely documented nontrading effect (French(1980)). Though evidence of the effect goes back to the early 1900s (Fields(1931)), authors such as Connolly (1989) and Chang, Pinegar, and Ravichan-dran (1993) show that it started to dissipate in the 1970s or 1980s. Chenand Singal (2003) hypothesize that the desire to close positions before theweekend would be most pronounced for short sellers who were deterred bythe unbounded downside risk inherent in their positions. When option mar-kets opened in the 1970s and 1980s, these investors were able to take bearishpositions with limited downside risk or hedge downside risk over nontrad-ing periods using options. Consistent with their hypothesis, Chen and Singal(2003) show that the introduction of options on a firm’s stock coincides with theelimination of the weekend effect for that firm.

Writing options also exposes a trader to unbounded downside risk, particu-larly over nontrading periods. For call options, this comes from the potentiallyinfinite positive payoff that accrues to the option buyer as the stock price rises.While delta-hedging reduces this exposure, the inability to perfectly hedge dueto transactions costs, price discontinuities, and nontrading periods means thatthis downside risk cannot be eliminated. That is, a call writer can experiencean arbitrarily large loss if the stock price rises high enough before the hedgecan be rebalanced. When writing put options, downside risk is bounded butoften extremely large, since the maximum payoff (which occurs when the stockprice drops to zero) can be orders of magnitude greater than the option’s price.Paradoxically, delta-hedging a put causes this downside to become unbounded,since hedging requires taking a short position in the stock, which itself isunbounded.

Given the above discussion, if investors are generally averse to holding posi-tions with extreme downside risk over the weekend, option writers may havean incentive to cover their positions by Friday close. While this should have noeffect in a Black and Scholes (1973) world in which options can be replicated viacontinuous trading, Bollen and Whaley (2004) show empirically that demandpressure does, in fact, have a substantial impact on option prices. Garleanu,Pedersen, and Poteshman (2009) add to this evidence and show theoretically

Option Mispricing around Nontrading Periods 863

that the effect is likely driven by the inability to perfectly replicate. Since cover-ing net written option positions can be regarded as positive demand pressure,any tendency to do so prior to Friday close should lead to temporarily higheroption prices that return to normal by the following Monday.

Alternatively, those same investors should be willing to keep their short po-sitions open given some additional compensation for the substantial downsiderisk they face. Since compensation to option writers comes in the form of nega-tive option returns, this argument suggests that we should also see a weekendeffect in call and put options, with weekend option returns being significantlylower than returns over the rest of the week.

Using a large sample of daily returns on equity options over the period fromJanuary 1996 to August 2014, we investigate nontrading effects in the equityoption market. We find pervasive, large, and highly statistically significant ev-idence that nontrading returns on options are lower than trading returns. Wefocus on average returns on delta-neutral positions, which are highly negativeover periods of nontrading (i.e., weekends and midweek holidays), but essen-tially zero on average on other days. There is no evidence of any nontradingeffect in underlying stock returns in our sample.

We find strong effects in puts and calls and across almost all levels of maturityand moneyness, with the results robust to different sampling methods andweighting schemes. The nontrading effect is negative in each year of our 19-year sample and is statistically significant in most. We also find strong evidenceof a nontrading effect in S&P 500 Index puts and weaker evidence of nontradingeffects in S&P 500 Index calls. We find significant nontrading effects for regularweekends, long weekends, and midweek holidays.

Nontrading effects are also evident in implied volatilities. As noted by Frenchand Roll (1986), the variance of the return over a weekend (Friday close toMonday close) is just slightly greater than the variance over a regular weekday,implying little variance during periods of market closure. However, impliedvolatilities seem to embed the expectation that stock price variance will remainsizable even when the market is closed. This discrepancy is large and significantfor both equity options and S&P 500 Index options.

We explore three possible explanations for these findings. The first is thatnontrading returns are lower because of differential levels of risk betweentrading and nontrading periods. The second is that aversion to unboundeddownside risk rises over nontrading periods, as hypothesized by Chen andSingal (2003). Since this risk is experienced by option writers but not buyers,option writers will require a higher risk premium to keep positions open. Sincewriters are short options, this means that option returns must be negative overthe weekend. The third explanation we consider is that the nontrading effectis the result of widespread and highly persistent market mispricing.

We find that portfolios of delta-neutral positions do show somewhat greaterrisk over nontrading periods relative to trading periods. For puts, the effectis modest: the put portfolio’s standard deviation is about 10% higher overnontrading periods (mostly weekends) than it is over regular trading intervals,and there is no significant change in return skewness or kurtosis. In contrast,


the call portfolio’s standard deviation is about 50% higher over nontradingperiods, and this is accompanied by a significant increase in kurtosis.

Despite the increase in risk over nontrading periods, writing options re-mains attractive according to a number of performance measures. Short optionpositions display positive and significant alphas over nontrading periods thatfar exceed those of trading periods. Sharpe ratios and information ratios arealso much higher over nontrading periods. We account for nonnormality in thedistribution of option returns by computing certainty-equivalent rates underpower utility and by computing the performance indexes of Kadan and Liu(2014), which are designed to account for the effects of higher moments. Foreach of the option portfolios considered, these measures are generally higherfor nontrading periods, in most cases by an amount that is both substantialand significant, with the strongest results for delta-hedged puts.

To examine whether the nontrading effect is associated with greater aversionto risk, rather than with risk by itself, we examine cross-sectional relationsbetween portfolio risk and the nontrading effect. We find that the magnitude ofthe effect is highly related to the Black and Scholes (1973) “gamma,” the secondderivative of the option price with respect to the stock price. We verify thatgammas have a strong positive association with the risk of delta-hedged optionportfolios, over both trading and nontrading periods. This is expected giventhat higher convexity reduces the effectiveness of delta-hedging, particularlywhen underlying price movements are large. Thus, if option writers have agreater aversion to downside risk over nontrading periods, as Chen and Singal’s(2003) hypothesis suggests, then higher gamma options should experience morenegative nontrading returns.

Likewise, the same logic would seemingly apply to “vega,” the derivative ofthe option price with respect to volatility. Higher vega makes an option moresensitive to changes in the volatility of the underlying asset and thereforeincreases the risk of the option position, even if it is delta-hedged. Indeed,we find that the risk in delta-hedged option positions is even more stronglyrelated to vega than it is to gamma. Surprisingly, however, greater vega doesnot seem to result in any additional return over nontrading intervals. Thus,greater risk aversion by option writers can only explain the nontrading effectgiven an explanation for why one type of risk exposure (gamma) is undesirable,while an even more important type (vega) is not.

The explanation that is more consistent with our findings is that investorsare pricing options using a method that does not properly account for thedifference in the behavior of stock prices when the market is open or closed.As French (1984) demonstrated over 30 years ago, if market participants donot account for the deterministic link between trading time and volatility, thenoption implied volatilities will tend to be too high just prior to nontradingperiods. Furthermore, option prices will embed a rate of time decay that is toolarge over nontrading periods and too small over trading periods. This wouldresult in a nontrading effect like the one we document, in particular, one that isdriven by gamma risk but not vega risk, as it is gamma rather than vega thatis closely tied to time decay. The consequence is widespread mispricing in the


option market, with abnormal rates of return on all options—but particularlythose with high gamma—negative over weekends and midweek holidays.

Our results imply that studies analyzing equity options data would benefitfrom accounting for nontrading effects, either by controlling for them explicitlyor by analyzing data at a frequency (e.g., weekly or intraday) at which thenontrading effect can safely be ignored. Given that the effect of nontrading onexpected returns is at least as large as the risk premia generated by traditionaloption pricing models, our findings also suggest that mispricing may play alarger role in the option market than has been emphasized in the literaturethus far. Since option prices are used in many types of research (volatilityestimation, reaction to corporate events, and understanding the risk-returnrelation) to infer the representative investor’s information set, we believe thatour findings should be of broad interest.

While a stand-alone trading strategy that captures the entire nontradingeffect would be difficult to achieve due to transactions costs, our results remainuseful for improving option-based trading strategies. Most obviously, nontrad-ing effects have clear implications for when option traders should enter and exittheir positions. Our findings should also be useful in setting quotes, whetherby market makers or other traders using limit orders.

On a practical level, our study suggests that market efficiency may be im-proved by greater awareness of the problems associated with option pricing incalendar time. Currently, option pricing models are often implemented in cal-endar time, rather than in trading time, both in archival data sets (e.g., IvyDB)and in market data feeds (e.g., the VIX Index). We have seen little discussionof the shortcomings of this approach. If market participants (traders and dataproviders) were to begin implementing models in trading time, we believe thatmarket efficiency could be improved measurably.1

We know of only one other paper that investigates weekly patterns in optionsmarkets. In examining short-term at-the-money options on 30 stocks over a pe-riod of just 21 months, Sheikh and Ronn (1994) find some evidence of a weeklyseasonal pattern in which call returns are highest on Wednesdays, but they donot find a weekend effect in calls or puts. After adjusting option returns by sub-tracting Black and Scholes (1973) model-implied returns, the pattern in callsdisappears, though a significant weekend effect arises in adjusted put returns.Our paper resolves the ambiguity of their findings, showing that weekend ef-fects are present in both calls and puts over a far more extensive sample.

The paper is organized as follows. In Section I, we describe our data, sam-pling methods, and portfolio construction. Section II contains our main resultsdocumenting nontrading effects in returns and implied volatilities. Section IIIexamines the relationship between nontrading effects and various measures ofrisk. Section IV concludes.

1 As French (1984) discusses, option pricing in trading time is also incorrect given that interestaccrues on a calendar-time basis. When interest rates are low, the use of trading time is highlypreferable even if imperfect.


I. Data and Methods

Our primary data source is the IvyDB data set from OptionMetrics. This dataset includes all U.S.-listed options on equities, indexes, and exchange tradedfunds. In our paper, we generally restrict attention to individual equity optionson stocks that were members of the S&P 500 as of the end of the prior year,though for some results, we examine all equity options or options on the S&P500 Index itself.

Throughout our analysis, we form portfolios of call and put options by takingthe equal- or value-weighted average of the returns, either hedged or unhedged,of each option in that portfolio. Our data set includes closing bid-ask quotesrather than transaction prices, so we compute values and returns from quotemidpoints. Since options have zero net supply, the concept of value-weightingmust be reinterpreted—what we call value-weighted portfolios are actuallyweighted by the dollar value of open interest for each option. Excess returnsare computed using the shortest maturity yield provided in the IvyDB zerocurve file.

In some cases, we form call and put portfolios on the basis of maturity and/ordelta. We consider three maturity ranges and six delta ranges so that all optionsin a given portfolio are roughly comparable and all portfolios have a reasonablenumber of options included. When we pool calls and puts into a single portfolio,calls and puts are each given a 50% weight, which is a way to make the resultingportfolio approximately delta-neutral and yet model-free.2

Because many option contracts are rarely traded, we choose our sample toalleviate the concern that our results are driven by untraded contracts. Wetherefore focus on a relatively liquid subsample, namely, options that havetraded with positive volume for five consecutive days as of the formation date.These contracts exhibit much higher trading volume on average during ourholding period, and they have much smaller bid-ask spreads.

Because individual equity options are American, early exercise can be opti-mal for in-the-money puts and for in-the-money calls that are about to pay div-idends, and the investment return based on optimal exercise could be greaterthan the no-exercise return we compute. To alleviate the problem for calls, weexclude all observations that occur in the five days leading up to and includ-ing a stock’s ex-dividend date, a conservative range that includes all dates onwhich early call exercise is optimal. Early exercise remains a minor issue fordeep-in-the-money puts.

We impose several additional filters to eliminate a small number of observa-tions that appear to be data errors, that violate arbitrage conditions, or thatappear to represent noncompetitive quotes. First, we eliminate observationsfor which there is a large reversal (2,000% followed by −95% or vice versa)in option returns or delta-hedged returns. We also eliminate options that vio-late arbitrage bounds. For options on equities, which are American, we require

2 In the Internet Appendix, which is available in the online version of the article on the Journalof Finance website, we consider an alternative portfolio in which calls and puts are matched onthe basis of maturity, delta, and open interest. We obtain similar results for this portfolio.


Table ISummary Statistics

This table reports basic summary statistics for our sample and for the larger data set from whichour sample is constructed. Panel A shows the average number of observations (contracts observed)per day, average percentage spread, and average volume (in terms of 100-share contracts) for fourdifferent samples, each of which is a subset of the previous one. The first sample is the full IvyDBdata set. The second only keeps options on firms that are members of the S&P 500. The third onlykeeps options with five consecutive days of positive trading volume. The final sample excludesobservations that fail one of the remaining filters described in Section I. Panel B reports momentsand betas for hedged and unhedged option portfolios constructed from the final sample, whereportfolio weights are proportional to lagged dollar open interest. Data are daily from January 4,1996 through August 28, 2014.

Panel A: Sample Characteristics

Full Data SetS&P 500

ConstituentsFive Days of

Positive Volume

Final Samplewith Additional

Filters

Average number of observations per dayPuts 61,860.0 19,588.5 1,797.9 1,635.0Calls 60,063.7 19,322.2 2,780.4 2,532.9

Average percentage bid-ask spreadPuts 52.06 51.38 15.15 14.56Calls 42.92 25.19 14.69 14.07

Average number of contracts tradedPuts 21.67 48.45 380.51 386.56Calls 35.91 77.34 435.84 422.28

Panel B: Return Moments

Mean SD Skewness Kurtosis Beta

Unhedged excess returnsPuts and calls −0.0023 0.0238 1.6841 10.8287 −0.4698Puts −0.0057 0.1060 0.7548 5.6799 −7.0761Calls 0.0011 0.0872 0.3924 4.9799 6.1364

Delta-hedged excess returnsPuts and calls −0.0012 0.0202 2.6735 29.7915 −0.9342Puts −0.0015 0.0204 2.1975 17.9576 −0.5344Calls −0.0008 0.0246 1.9672 51.2985 −1.3339

that the option price be no less than the current exercise value. For calls, theprice must be less than the current stock price. For puts, it must be less thanthe strike. We also eliminate observations for which the bid price exceeds theask price or the bid-ask spread is more than $10 or more than the price ofthe underlying stock, as these observations might be data recording errors,noncompetitive “stub” quotes posted by a market maker who does not want totrade, or undocumented missing value codes (e.g., 999). We also require, at thedate of portfolio formation, that the bid price be above zero and that the bid-askspread be no more than twice the bid.

The top panel of Table I contains statistics that describe the average crosssection in our sample. On average, the IvyDB data set includes around 60,000


calls and puts per day. About one-third of those are on S&P 500 stocks, andabout one-tenth are what we consider highly liquid. A small number of thoseobservations are filtered out, leaving us with around 1,800 puts and 2,800 callsper day. The contracts we include have relatively small bid-ask spreads, around15% on average, compared with 51% and 25% for puts and calls, respectively,for the entire S&P 500 sample. They also trade much more, at roughly 400contracts per day on average, which is six to eight times more than the fullS&P 500 sample.3

Taken together, the filters above eliminate the largest and most suspiciousoutliers in our sample and ensure that we focus primarily on more liquidoptions. We note that all of our main results are robust to dropping any or all ofthese filters, and that further tightening the requirements on trading volumeincreases the estimated magnitude of the nontrading effect. These robustnessresults are reported in Section II.B.

Our analysis requires the use of implied volatilities, and unless otherwisenoted, we use those provided by IvyDB, which are computed using a binomialtree approach that accounts for dividends. Because they are equivalent to Blackand Scholes (1973) values for stocks that pay no dividends, we refer to theimplied volatilities and “Greeks” as “Black-Scholes” values. The option pricesensitivities of each option, that is, delta, gamma, vega, and theta, are alsocomputed with the Black and Scholes (1973) model using the implied volatilityof the same option.

The IvyDB data set does not include an implied volatility for about 3% of theobservations included in our analysis. Following Duarte and Jones (2008), wefill in missing implied volatilities with those of similar contracts. Specifically, ifa call option’s implied volatility is missing, we use the implied volatility of theput contract written on the same underlying firm with the same maturity andstrike price. If both put and call implied volatilities are missing, we use thevalue from the same contract on the previous day. It is worth noting that thisprocedure for filling in missing implied volatilities relies only on current andlagged information. A portfolio strategy that uses the implied volatilities fromthis procedure for portfolio formation and to compute hedge ratios is thereforefully implementable. Furthermore, our results are robust to filtering out theseobservations.

We compute option returns based on closing bid-ask midpoints, and excessreturns account for the number of calendar days within the return holdingperiod.4 The change in value of a delta-hedged portfolio is

Ct − Ct−1 − �t (St − St−1) ,

3 Trading volumes reported in Table I are those during the holding period, while the filterrequiring five consecutive days of positive volume is applied to the five days prior to the holdingperiod. The spreads reported in the table are at the start of the holding period. Spreads on the endof the holding period are almost identical.

4 As highlighted by French (1984), interest accrues on a calendar-time basis.


where Ct is the option price, St is the stock price, and �t is the Black andScholes (1973) delta. Delta-hedged excess returns are therefore defined as

Ct − Ct−1Ct−1

− rt−1NDt−1,t − �tSt−1Ct−1

(St − St−1

St−1− rt−1NDt−1,t

),

where rt is the riskless return per day and NDt−1,t is the number of calendardays between date t − 1 and t. This may be viewed as the excess return ona portfolio that combines one option contract with a zero-cost position in �tshares worth of single-stock futures.

For each delta-hedged option portfolio, there is a corresponding hedge port-folio that consists of all the equity positions taken to eliminate delta. Thisportfolio has weights proportional to

−�tSt−1Ct−1

,

which is negative for calls and positive for puts. We examine these portfoliosbriefly to rule out the possibility that nontrading effects arise from the under-lying stock returns.

The bottom panel of Table I reports moments on value-weighted portfoliosof puts and calls, both separately and combined in a 50/50 portfolio. Sev-eral stylized facts are immediately apparent. First, portfolios of unhedgedcall options have positive average returns, while unhedged puts have nega-tive average returns, which is consistent with calls having positive marketbetas and puts having negative betas. Delta-hedging makes all average re-turns negative, consistent with the vast literature (e.g., Coval and Shumway(2001) and Bakshi and Kapadia (2003)) documenting a negative variance riskpremium.

A second observation is that delta-hedging dramatically reduces the stan-dard deviations and betas of put and call portfolios. Delta-hedging reduces thestandard deviation of the put portfolio by over 80% and reduces the beta ofthat portfolio from −7.08 to −0.53. Slightly smaller reductions are observed forthe call portfolio. In contrast, little reduction in standard deviation is observedfor the combined portfolio of puts and calls, and the beta of that portfolio isactually larger in absolute value as a result of hedging. This is because a 50/50portfolio of puts and calls is already approximately delta-neutral, so additionaldelta-hedging is not particularly effective.

Third, portfolios of unhedged option positions have returns that display posi-tive skewness and substantial excess kurtosis. Delta-hedged option returns areeven more fat-tailed, the result of delta-hedging being more effective for smallchanges in stock prices. Large changes in stock prices, which are often thesource of extreme option returns, cannot be hedged due to the convexity of op-tion payoffs. Tails thicken as a result of a relative decrease in the low-volatilityreturns, for which delta-hedging is effective.

In the remainder of our paper, we focus on the delta-hedged call portfolio,the delta-hedged put portfolio, and the unhedged 50/50 portfolio of puts and


calls. Because all three of these portfolios are approximately delta-neutral, ouranalysis focuses on the component of option prices that is orthogonal to theunderlying stocks. This is desirable given that prior literature shows that thenontrading effect has vanished in equities during our sample period, a resultthat we confirm below in our own sample.

We view our use of the Black and Scholes (1973) model as relatively benign.Even if the delta we use to compute hedged returns is somewhat misspecified,the hedged returns nevertheless represent the returns on a feasible investmentstrategy (abstracting from transaction costs). Delta-hedging, while not perfect,can be expected to remove at least the majority of an option’s exposure to theunderlying stock. Indeeed, Hull and Suo (2002) find that Black-Scholes workabout as well as any other model in this regard. Any remaining concerns shouldbe lessened when we combine unhedged calls and puts into a single portfolioin a way that is model-free.

II. Nontrading Effects in Option Markets

A. Main Results

Our main finding is that returns in delta-neutral option positions are signif-icantly lower over nontrading periods than trading periods. We define a non-trading period as any period in which the interval between subsequent closingprices is longer than one calendar day. The vast majority of these nontradingperiods are regular weekends, with a smaller number consisting of midweekholidays and long weekends. We begin by pooling all types of nontrading periodstogether. Later, we examine them separately.

We present our main results in Tables II and III. Table II reports resultsfor value-weighted portfolios of all puts and all calls, while Table III reportsresults for value-weighted portfolios of puts and calls formed on the basis ofmaturity and delta.

When we examine the unhedged put-call portfolio in Table II, we observe anaverage nontrading excess return of −0.76% and an average trading return of−0.08%. The difference between the two, which we define as the nontradingeffect, is equal to −0.68% and under any convention is highly significant.5 Fordelta-hedged puts, the average nontrading excess return is −0.82%, relativeto an average trading return of 0.03%. The difference, −0.86%, is even morehighly significant. For calls, the difference between nontrading and tradingreturns is −0.51% and again statistically significant. The smaller effect forcalls relative to puts is notable but is driven largely by the fact that put and

5 Here and elsewhere in the paper, standard errors for mean and difference-in-mean estimatesare obtained by regressing returns on a constant and a nontrading dummy, then applying theNewey-West (1987) procedure with 22 lags to account for possible autocorrelation. We have alsocomputed p-values for all of our key results using the block bootstrap, and we find these results tobe very close.


Tab

leII

Ave

rage

Ret

urn

sfo

rT

rad

ing

and

Non

trad

ing

Per

iod

san

dfo

rE

ach

Day

ofth

eW

eek

Th

ista

ble

repo

rts

aver

age

exce

ssre

turn

sof

port

foli

osof

un

hed

ged

and

delt

a-h

edge

deq

uit

yop

tion

s.T

he

firs

tth

ree

colu

mn

sof

the

tabl

esh

owcl

ose-

to-c

lose

retu

rns

intr

adin

gan

dn

ontr

adin

gpe

riod

sas

wel

las

the

diff

eren

cebe

twee

nth

em.T

he

rem

ain

ing

colu

mn

ssh

owav

erag

ere

turn

sby

day

ofth

ew

eek.

Th

eu

nh

edge

dpu

tan

dca

llpo

rtfo

lio

isco

nst

ruct

edas

the

50/5

0av

erag

eof

put

and

call

port

foli

osth

atar

eea

chw

eigh

ted

acco

rdin

gto

lagg

eddo

llar

open

inte

rest

.Hed

ged

puts

and

hed

ged

call

sar

epo

rtfo

lios

ofde

lta-

hed

ged

opti

ons,

wei

ghte

dac

cord

ing

tola

gged

doll

arop

enin

tere

st.T

he

stoc

kh

edge

port

foli

osfo

rpu

tsan

dca

lls

are

the

port

foli

osof

stoc

ksth

atar

eu

sed

tode

lta-

hed

geth

epu

tan

dca

llop

tion

s,re

spec

tive

ly.V

alu

esin

pare

nth

eses

are

t-st

atis

tics

com

pute

dfr

omN

ewey

-Wes

t(1

987)

stan

dard

erro

rsw

ith

22la

gs.D

ata

are

dail

yfr

omJa

nu

ary

4,19

96th

rou

ghA

ugu

st28

,201

4.

Trad

ing

Non

trad

ing

Dif

fere

nce

Mon

day

Tues

day

Wed

nes

day

Th

urs

day

Fri

day

Un

hed

ged

puts

and

call

s−0

.000

8−0

.007

6−0

.006

8−0

.007

9−0

.002

6−0

.000

80.

0017

−0.0

021

(−1.

76)

(−9.

30)

(−7.

86)

(−8.

18)

(−4.

16)

(−1.

14)

(1.8

8)(−

2.81

)H

edge

dpu

ts0.

0003

−0.0

082

−0.0

086

−0.0

085

−0.0

016

0.00

190.

0025

−0.0

023

(0.8

1)(−

11.9

1)(−

11.9

3)(−

10.3

7)(−

2.78

)(2

.72)

(3.3

0)(−

2.99

)H

edge

dca

lls

0.00

03−0

.004

8−0

.005

1−0

.004

9−0

.002

20.

0012

0.00

31−0

.001

7(0

.69)

(−5.

08)

(−5.

11)

(−5.

34)

(−2.

69)

(1.2

6)(3

.46)

(−2.

38)

Sto

ckh

edge

for

puts

0.00

370.

0058

0.00

200.

0045

0.00

730.

0084

0.00

18−0

.001

2(2

.25)

(1.8

4)(0

.55)

(1.2

5)(2

.28)

(2.4

7)(0

.56)

(−0.

43)

Sto

ckh

edge

for

call

s−0

.001

5−0

.003

6−0

.002

0−0

.002

1−0

.005

9−0

.003

70.

0005

0.00

15(−

0.97

)(−

1.11

)(−

0.55

)(−

0.59

)(−

1.72

)(−

1.04

)(0

.17)

(0.5

2)U

nh

edge

dpu

ts−0

.003

4−0

.014

0−0

.010

6−0

.013

0−0

.008

9−0

.006

60.

0007

−0.0

011

(−1.

88)

(−4.

04)

(−2.

72)

(−3.

18)

(−2.

71)

(−1.

79)

(0.2

1)(−

0.34

)U

nh

edge

dca

lls

0.00

18−0

.001

3−0

.003

1−0

.002

80.

0037

0.00

490.

0026

−0.0

032

(1.3

0)(−

0.47

)(−

0.97

)(−

0.89

)(1

.23)

(1.6

9)(1

.06)

(−1.

30)


Table IIIDifferences between Returns Following Nontrading Days and

Trading Days for Portfolios Sorted by Delta and MaturityThis table reports the average excess returns on nontrading days minus the average excess returnson trading days for single- and double-sorted portfolios formed on the basis of delta and maturity.Corresponding Newey-West (1987) t-statistics are also shown in parentheses. Portfolios of bothunhedged and delta-hedged equity options are considered, though the former are only single-sorted by maturity. All methods are identical to Table II except for the classification into moredisaggregated portfolios.

All 11 to 53 54 to 118 119 to 252Maturities Days Days Days

Unhedged puts and callsAll deltas −0.0068 −0.0103 −0.0059 −0.0026

(−7.86) (−9.55) (−8.25) (−3.34)Hedged puts

All deltas −0.0086 −0.0117 −0.0069 −0.0043(−11.93) (−12.66) (−11.34) (−7.06)

−0.90 > delta ≥ −0.99 −0.0020 −0.0020 −0.0007 −0.0007(−5.90) (−5.51) (−2.09) (−1.93)

−0.75 > delta ≥ −0.90 −0.0040 −0.0041 −0.0020 −0.0015(−9.39) (−9.28) (−4.05) (−3.82)

−0.50 > delta ≥ −0.75 −0.0066 −0.0083 −0.0047 −0.0026(−10.81) (−11.50) (−9.43) (−5.50)

−0.25 > delta ≥ −0.50 −0.0104 −0.0158 −0.0080 −0.0049(−11.75) (−11.60) (−11.62) (−7.01)

−0.10 > delta ≥ −0.25 −0.0173 −0.0273 −0.0121 −0.0074(−10.59) (−10.21) (−10.03) (−6.96)

−0.01 > delta ≥ −0.10 −0.0245 −0.0328 −0.0163 −0.0065(−7.04) (−7.30) (−6.20) (−2.82)

Hedged callsAll deltas −0.0051 −0.0076 −0.0051 −0.0031

(−5.11) (−5.57) (−5.57) (−3.44)0.01 < delta ≤ 0.10 −0.0358 −0.0436 −0.0151 −0.0160

(−4.94) (−5.56) (−2.70) (−2.85)0.10 < delta ≤ 0.25 −0.0230 −0.0312 −0.0162 −0.0097

(−7.08) (−7.48) (−6.44) (−3.28)0.25 < delta ≤ 0.50 −0.0106 −0.0159 −0.0087 −0.0051

(−7.25) (−7.54) (−7.15) (−4.50)0.50 < delta ≤ 0.75 −0.0043 −0.0065 −0.0034 −0.0023

(−5.63) (−5.93) (−5.43) (−3.60)0.75 < delta ≤ 0.90 −0.0014 −0.0017 −0.0011 0.0000

(−3.19) (−3.02) (−2.96) (−0.01)0.90 < delta ≤ 0.99 0.0000 −0.0002 0.0001 −0.0001

(−0.06) (−0.59) (0.25) (−0.35)

call portfolios weight in-the-money and out-of-the-money options differently, aswe show below.6

6 Even in an equal-weighted portfolio, differences between average call and put returns arepossible and do not suggest any violation of put-call parity, which would imply similar effects onthe prices of puts and calls rather than on their returns.


To gauge the extent to which the nontrading portfolio is present in the un-derlying stocks, Table II reports average excess returns over trading and non-trading periods for the portfolios of underlying stocks used to delta-hedge theput and call portfolios. The put portfolio is hedged by going long the underlyingstocks, and we see that the long stock portfolio experienced somewhat higherreturns during nontrading periods, though the difference is not significant.Calls are hedged by shorting stocks, so the hedge portfolio for calls earnedlower returns over nontrading periods. Given that the difference between trad-ing and nontrading returns in stocks is insignificant and that any differencehas opposite effects on puts and calls, it is unlikely that the nontrading effectin options is driven by a nontrading effect in stocks.

Table II also reports averages of unhedged put and call portfolios separately.From Table I, we know that these portfolios are several times more volatilethan the corresponding hedged portfolios given their large betas. Because ofthis, the nontrading effect is much more difficult to detect and is statisticallysignificant only in puts.

The right side of the table summarizes average returns by day of the week,where the day given is the day at the end of the holding period. Mondays natu-rally represent the vast majority of nontrading-period ends, with Tuesdays andFridays being the next most frequent (typically following three-day weekendsand Thanksgiving holidays, respectively). Overall, delta-hedged excess returnsare lowest on Mondays, consistent with previous results. Smaller negative re-turns on other days are consistent with a variance risk premium, though thepositive average returns observed on Wednesdays and Thursdays are some-what anomalous. In the Internet Appendix, we show that robust estimators,which downweight large outliers, eliminate these positive average returns,while the nontrading effect is robust to a variety of alternative estimators andto the use of block bootstrap standard errors.

Table III examines the nontrading effect, or the difference between non-trading and trading returns, for disaggregated portfolios. We form unhedgedcall/put portfolios on the basis of maturity, measured as the number of tradingdays until option expiration, and delta-hedged call and put portfolios based onmaturity and moneyness, where we use lagged option deltas to measure mon-eyness.7 In 59 of the 60 portfolios, average nontrading returns are less thanaverage trading returns, with 54 of these differences statistically significant.One difference is positive but insignificant.

Several patterns emerge from the table. First, the nontrading effect isstronger for shorter term options, being several times higher for options inthe 11- to 53-day range relative to the 119- to 252-day range. Second, the effectis much larger for out-of-the-money options (deltas close to zero) than it is forin-the-money options (deltas close to −1 or +1). Both of these patterns suggesta possible relation between the nontrading effect and one or more of the Black

7 We do not double-sort the unhedged put-call portfolio because this would result in portfolioswith large negative or positive deltas whose returns were dominated by their underlying stockexposures.


Figure 1. Average unhedged put and call portfolio returns in each calendar year. Thisgraph depicts the average return on the unhedged put and call portfolio in each year of our sample.The solid line denotes the average, and the two dashed lines denote the upper and lower bounds ofa 95% confidence interval constructed using Newey-West (1987) standard errors. The shaded areashows the average three-month LIBOR rate within each calendar year.

and Scholes (1973) Greeks, which measure the sensitivity of option values tovarious factors and are highly affected by maturity and moneyness. We returnto this issue in Section III.C, where we directly examine the relation betweenoption characteristics and the nontrading effect.

A final result from Table III is that, for the same absolute delta, which is ameasure of option moneyness, there is little tendency for the nontrading effectin puts to exceed that of calls. This confirms that the greater nontrading effectfound for the aggregated put portfolio in Table II was primarily a compositioneffect, driven by the fact that open interest in out-of-the-money options is largerfor puts than calls.

To gauge the consistency of our results, we recompute the nontrading effectfor each year of our sample. Figure 1 plots the yearly estimates for the unhedgedcall/put portfolio along with asymptotic 95% confidence intervals. In each of the19 years of our sample, the nontrading effect is negative. In 11 out of the 19years, the effect is statistically significant. We observe similar results in thedelta-hedged portfolios, where 18 out of 19 years are negative for puts and 16out of 19 are negative for calls.

We do see that the nontrading effect in the second half of the sample issubstantially more volatile and slightly smaller on average relative to thatin the first half of the sample. Some of these changes are likely driven bythe extreme volatility of the recent financial crisis. It is also possible that theattenuation of the effect is related to the rise of algorithmic trading and thereplacement of traditional market makers by high-frequency trading firms,


both of which followed the 2003 creation of Options Linkage, which connectedall option exchanges in the United States (see Muravyev and Pearson (2016)).It is notable, however, that the last year of the sample displays an above-average effect, indicating that the effect has survived these changes to marketstructure.

The figure also shows the average three-month London Interbank OfferedRate (LIBOR) in each year of the sample to alleviate the concern that non-trading effects arise through an interest rate channel. Ex ante, such an effectis unlikely for several reasons. First, our returns are in excess of the risklessrate, with the correct adjustment for the number of calendar days in the hold-ing period. Any residual effect of interest rates can therefore arise only throughrisk premia. One might be concerned, for instance, that unhedged risks (vegaand gamma) become larger over nontrading periods when interest rates arehigh. However, at least under the Black and Scholes (1973) model, the effectsof interest rates on Greeks such as vega and gamma are small for most options,and negligible for options with one month till expiration, where our results arestrongest. Second, prior literature, including French (1984), Scott (1997), andBakshi, Cao, and Chen (1997, 2015), generally finds a limited role for interestrates in the pricing of equity or equity index options, in that the precise speci-fication of the interest rate process has little effect on option prices or returns.In any case, it is evident from Figure 1 that there is no relation between thetwo series plotted, which confirms that nontrading effects do not arise throughan interest rate channel.

As discussed in Section I, a potential concern is that our returns are cal-culated assuming no early exercise. Since we eliminated observations on andimmediately prior to dividend ex-days, early exercise will never be optimal forcalls, though it could be optimal for some in-the-money puts. For such puts,the value of early exercise comes from accelerating the fixed option premiumforward in time. While this can have a significant effect on option value, theloss in value that results from delaying exercise by one day should be small, asit is at most the loss of one day of interest on the strike price. Thus, for theseoptions, our computed returns should at most slightly understate the returnsof a strategy in which exercise decisions are made optimally.

More importantly, early exercise cannot explain our results because it isnever optimal for call and put options that are out of the money. Since wefind large nontrading effects across all moneyness levels, early exercise due todividends or any other factor cannot explain our findings.

B. Robustness to Portfolio Construction and Subsamples

In Table IV, we examine nontrading effects in different samples to determinewhether our main results are sensitive to the choice of empirical approach.Panel A reports results using a number of alternative methods for computingoption portfolio returns. We first replicate our main results using ask-to-bidand bid-to-ask returns in addition to the midpoint-based returns that we con-sider throughout the paper. All three methods for calculating returns lead


Table IVNontrading Effects in Other Portfolios and Samples

This table reports nontrading effects, defined as a nontrading mean minus a trading mean, foralternative portfolio constructions and for changes in the implied volatility of at-the-money options.Unless noted otherwise, all portfolios are constructed from the main sample (S&P 500 stocks, fivedays of positive volume); are weighted by dollar open interest; are based on quote midpoints;and impose filters for dividends, arbitrage violations, price reversals, and maximum spreads.In Panel A, we compute option returns as either midpoint-to-midpoint, ask-to-bid, or bid-to-ask.Results based on midpoints are identical to those in Table II. We also compute equal-weightedportfolios and portfolios weighted by open interest measured in terms of contracts rather thandollars. Panel B shows results for the following alternative data samples: options on all stocks (notjust S&P 500 stocks, no trading volume requirement); the main sample but without filters; a morerestricted sample of options with five consecutive days of trading 100 or more contracts; and a morerestricted sample of options with relative bid-ask spreads of 5% or less. Panel C shows results forS&P 500 Index options, including deep-out-of-the-money options with Black-Scholes deltas below0.1 in absolute value. Panel D reports nontrading effects for different types of nontrading periods,namely, regular weekends, midweek holidays, and long weekends with three or more days ofmarket closure. Newey-West (1987) t-statistics are in parentheses. Data are daily from January 4,1996 through August 28, 2014.

Panel A: Nontrading Effects Using Alternative Return Constructions and Portfolio Weights

Dollar Open Interest Weighted

Midpoints Ask to Bid Bid to AskEqual

WeightedContractWeighted

Unhedged puts and calls −0.0068 −0.0060 −0.0079 −0.0197 −0.0184(−7.86) (6.99) (8.38) (−14.75) (−13.32)

Hedged puts −0.0086 −0.0075 −0.0100 −0.0213 −0.0173(−11.93) (9.95) (11.02) (−14.54) (−11.06)

Hedged calls −0.0051 −0.0045 −0.0059 −0.0162 −0.0160(−5.11) (3.98) (5.91) (−7.25) (−5.80)

Panel B: Alternative Data Samples

Main Sample

Options onAll Stocks

Zero VolumeContracts

Only No FiltersFive Days ofVolume>100

Maximum5% Spread

Unhedged puts and calls −0.0043 −0.0027 −0.0072 −0.0101 −0.0044(−8.50) (−8.08) (−8.52) (−6.72) (−4.69)

Hedged puts −0.0061 −0.0042 −0.0085 −0.0112 −0.0063(−15.38) (−16.31) (−12.45) (−8.42) (−9.24)

Hedged calls −0.0037 −0.0018 −0.0053 −0.0074 −0.0029(−5.32) (−4.49) (−5.04) (−4.73) (−4.00)

Panel C: Nontrading Effects in S&P 500 Index Options

Value Weighted Equal Weighted Equal Weighted, Deep OTM

Puts and calls −0.0073 −0.0250 −0.0471(−3.87) (−4.93) (−4.70)

Puts −0.0107 −0.0315 −0.0519(−4.56) (−5.34) (−4.95)

Calls −0.0030 −0.0155 −0.0453(−0.91) (−1.66) (−2.41)

(Continued)


Table IV—Continued

Panel D: Nontrading Effects in Different Nontrading Periods

Regular Midweek LongWeekends (A) Holidays (B) Weekends (C) (B)−(A) (C)−(A)

Puts and calls −0.0079 −0.0032 −0.0071 0.0048 0.0009(−8.06) (−0.83) (−3.39) (1.21) (0.40)

Puts −0.0087 −0.0062 −0.0056 0.0025 0.0031(−10.20) (−2.96) (−2.79) (1.04) (1.45)

Calls −0.0052 −0.0065 −0.0016 −0.0013 0.0036(−5.03) (−1.82) (−0.53) (−0.33) (1.11)

Number of observations 851 44 122

to similar conclusions about the strength and significance of the nontradingeffect.

We next compute portfolio returns using equal weighting instead of weight-ing by the dollar value of open interest. Equal weighting has several effects.First, it puts relatively more weight on less liquid contracts that do not tradeas frequently and that hold little open interest. Second, it likely increases themeasurement error bias identified by Blume and Stambaugh (1983), thoughthis should affect both trading and nontrading returns and hence possibly washout when computing their difference. Finally, and most importantly, it puts rel-atively more emphasis on lower priced contracts, which are typically shorterterm and deeper out of the money. Since we previously found that the nontrad-ing effect is most pronounced for short-term deep-out-of-the-money contracts,equal weighting should increase the size of the nontrading effect. Consistentwith this conjecture, we find that equal weighting causes the nontrading effectto more than double.

When we instead weight by the open interest in terms of the number of con-tracts rather than the dollar value, we obtain results that are very similar tothose for the equal-weighted portfolios. This suggests that the larger nontrad-ing effects in the equal-weighted portfolio are not a result of overweightingthinly traded options, but are instead due to the equal-weighted portfolio’sgreater emphasis on low-priced deep out-of-the-money contracts.

In Panel B of Table IV, we consider different option samples. We start byexamining options on all U.S.-listed equities rather than limiting attentionto those that are members of the S&P 500, and we do not require that con-tracts trade for five consecutive days. The resulting sample is much larger butsubstantially less liquid. In this sample, the nontrading effect is reduced byroughly one-third across the three portfolios, but it remains highly statisticallysignificant. To see if the effect holds in even the most illiquid contracts, wefurther reduce liquidity by restricting the sample to include only those optionswith zero trading volume on the portfolio formation day. The effect is againslightly smaller but still highly significant.

We also examine whether our use of filters (dividends, reversals, etc.) has anyeffect on our results. We find that dropping these filters has almost no effect on


our main results, though they are likely more important when we analyze lessaggregated portfolios.

In addition, we examine whether samples that are even more liquid than theone we focus on also show nontrading effects. We construct one such sampleby requiring option contracts to have five consecutive days of trading at least100 contracts, which reduces the sample by about 85% relative to our primarysample (S&P 500 firms, five days of positive volume).8 The result is that thenontrading effect is amplified, roughly 40% larger for each of the three portfoliosconsidered. Overall, it appears that the effect we document in this paper isperhaps twice as large for the most heavily traded options as it is for the leasttraded ones.

We also analyze a subsample of options with bid-ask spreads no greater than5% of the quote midpoint, which should further minimize concerns regardingour reliance on bid-ask midpoints in computing returns. For these portfolios,the size of the nontrading effect is about two-thirds that of the baseline portfo-lios, which would appear to go against other results suggesting that the effect islarger in more liquid contracts. The explanation is that low-percentage spreadsare more common among in-the-money contracts, where the nontrading effectis weaker. Liquidity, in this case, is proxying for moneyness.

Panel C examines the presence of a nontrading effect in S&P 500 Indexoptions, where we impose all filters used to construct our primary sample exceptfor the one eliminating dividends. Intuitively, one might expect index optionsto be driven by the same systematic risk factors that affect individual equityoptions. However, the two differ significantly in that index options are optionson a portfolio rather than a portfolio of options. They are therefore affected bycorrelations as well as volatilities. Risk factors that drive correlations, as in themodel of Driessen, Maenhout, and Vilkov (2009), should therefore affect onlyoptions on the index.

In contrast, systematic movements in average idiosyncratic volatility willaffect portfolios of individual options but will have little effect on options onthe index. Such systematic movements have been documented by Campbellet al. (2001) and have been found by Goyal and Santa-Clara (2003) to driveaggregate risk premia. If the risk premium on this factor has a weekly seasonalpattern, then this will lead to a weekend effect only in individual equity options.

Table IV shows that the average nontrading effects for the value-weightedunhedged put-call portfolio and the delta-hedged put portfolio are both slightlylarger than the effects measured from equity options, though the t-statistics aresomewhat lower. We find no significant effect in delta-hedged calls, though thepoint estimate in index call options is negative and sizable. The reason for theweak results on index calls appears to be related to differences in the levels ofopen interest of calls and puts across various levels of maturity and moneyness.

8 Requiring 1,000 contracts per day reduces the sample much further still. Doing so has littleimpact on the nontrading effect, raising the effect slightly for calls and lowering it slightly forputs. However, standard errors are larger due to a smaller sample, reducing the significance of theeffect.


For example, deep-out-of-the-money puts (deltas greater than −0.1), which arecommonly used to obtain portfolio insurance against market risk, compriseabout 3% of the total dollar value of puts traded, on average. Deep-out-of-the-money calls (deltas below 0.1) comprise only 0.6% of the total amount of calls.Since out-of-the-money contracts exhibit the largest nontrading effects, theaverage nontrading effect for calls is reduced.9

If we instead form portfolios that are equal weighted, which gives relativelymore weight to out-of-the-money contracts, all nontrading effects rise substan-tially, but the effect is still insignificant for S&P 500 Index calls. However,if we focus solely on calls with deltas less than 0.1, the effect is statisticallysignificant and, at around −4.5%, extremely large in economic magnitude.10

Panel D of Table IV reports estimates of the nontrading effect for differenttypes of nontrading periods, all of which have been included in our analysisthus far. For each type, the values reported are the average returns on the non-trading period described (e.g., midweek holidays) minus the average returnsover all trading days. If what we are documenting is truly a nontrading effectrather than a weekend effect, then we should also see negative option returnsover midweek holidays, and we might also expect the effect to be stronger overlong weekends.

In our sample, there are only 44 midweek holidays, so the average returnsover these periods are estimated imprecisely. Nevertheless, this small sub-sample shows a highly significant nontrading effect for hedged puts and amarginally significant result for hedged calls, which supports the view thatour findings represent a nontrading effect rather than weekend effect. On theother hand, the nontrading effect for unhedged options in this small sampleis negative but insignificant. The table also reports results on the differencebetween the nontrading effect over midweek holidays and that of regular week-ends. No significant difference is observed, but this may again be attributableto the small sample of midweek holidays.

Long weekends, with three or more days of nontrading, are somewhat morecommon. During these periods, we find significant nontrading effects for un-hedged options and hedged puts, though not for hedged calls. As with midweekholidays, there is no significant difference between the nontrading effect overlong weekends and that of regular weekends. Still, the effects appear weakerrather than stronger over long weekends relative to regular weekends. Whilethe weaker nontrading effect over these periods should be taken with a grainof salt given the small number of long weekends, it may also be the case thatsome options traders are aware of the nontrading effect and try to write optionsto benefit from it in more conspicuous cases, such as long weekends. As indirect

9 We explore the relation between option moneyness and the strength of the nontrading effectin Section III.C, where we provide evidence that the relation is due to a greater amount of excesstime decay in out-of-the-money options.

10 The Internet Appendix examines S&P 500 Index options in more detail. In short, the evidenceis statistically weaker than that for the equity option sample that we focus on in the paper, butpoint estimates of the effect for different maturities and deltas are generally similar to those foundin equity options.


evidence of this interpretation, we find that average hedged put and call re-turns are −0.53% and −0.37%, respectively, on the day before a long weekend,compared to just −0.04% and −0.07% on the day before a regular weekend.These differences are statistically significant at the 5% level.

Taken together, the results in Table IV show that the nontrading effect isextremely robust but more pronounced in liquid contracts. While the effect isinsignificant in S&P 500 Index calls, it is large and significant in puts. Thisis notable because the markets for index and equity options are different inmany respects. For instance, as Garleanu, Pedersen, and Poteshman (2009)observe, the “end users” presumed responsible for variation in option demandare on average buyers of index options but writers of equity options. The factthat both types of options exhibit nontrading effects of the same sign andsimilar magnitude appears to challenge demand-based option pricing as anexplanation for our findings. Furthermore, we find a strong effect even whenthere is no apparent change in demand, namely, in contracts with zero tradingvolume. This does not necessarily rule out the demand-based option pricingmodel as an explanation for the nontrading effect, considering that Garleanu,Pedersen, and Poteshman show that option prices should be affected by tradesin other contracts. Nevertheless, the strength of the effect in untraded contractsis surprising.

C. Implied and Realized Variances

In this section, we assess the significance of nontrading effects in impliedvariances. Specifically, we ask whether the variance of returns over nontradingperiods that is implicit in option prices matches the realized variance from stockreturns.

Define the quadratic variation between times u and v as

QVu,v =∫ v

s=uσ 2s ds,

where σt is the instantaneous volatility of the log stock price process. Nowdefine the time-t implied variance of an option with fixed expiration date T as

IVt,T = EQt[QVt,T

] = EQt [QVt,t′ + QVt′,T ] .Note that we are measuring the total amount of variance remaining over theoption’s lifetime rather than dividing by the time to expiration, which wouldresult in a variance rate.

The change, from time t to time t′, in implied variance remaining until expi-ration is

IVt′,T − IVt,T = EQt′[QVt′,T

] − EQt [QVt,T ] .This change is on average negative as a result of moving closer to expiration,as the quadratic variation between times t and t′ is eliminated from impliedvariance at time t′. In studies of earnings announcements, Patell and Wolfson


(1981) and Dubinsky and Johannes (2006) use this change to estimate theannouncement-day variance for a sample of stocks.

If we add the realized quadratic variation of the underlying asset over thesame period to offset the effect of moving closer to expiration, we get a variance“differential,”

Dt,t′ ≡ IVt′,T − IVt,T + QVt,t′ , (1)which measures the difference between the actual variance and the declinein implied variance remaining until expiration. This differential can be reex-pressed as

Dt,t′ =(EQt′

[QVt′,T

] − EQt [QVt′,T ])

+(QVt,t′ − EQt

[QVt,t′

]). (2)

The first term represents the change in the risk-neutral expectation of futurevariance over the interval (t′, T ), which is equal to the price appreciation on aforward start variance swap. Without a variance risk premium, forward pricesare martingales, so this term should be zero in expectation. The second termrepresents the difference between actual and implied variance. Again, the ab-sence of a variance risk premium should make this term zero in expectation.Thus, if there is no variance risk premium, that is, if implied variance is an un-biased forecast of future-realized variance, then the expectation of Dt,t′ shouldbe zero. This statement holds regardless of the length of time between t and t′

and regardless of whether that interval includes a nontrading period.To determine whether options embed an expectation of future variance over

nontrading periods that is larger than the actual variance over nontradingperiods, we measure the variance differential Dt,t+1 for each stock and for theS&P 500 Index on each day in our sample. For the equity option sample, weaverage these values across stocks to obtain a single value on each day. If thenontrading variance implicit in option prices exceeds that present in actualreturns, then Dt,t+1 should be negative on average over nontrading periods.

We measure the variance differential using two different methods for comput-ing implied variances, which are based on either the Black and Scholes (1973)formula applied to at-the-money options or model-free implied variances. Inthe former case, the implied variance is the average of the values computedfrom the call and put with deltas closest to 0.5 and −0.5, respectively. In thelatter case, we apply the interpolation method of Hansis, Schlag, and Vilkov(2010) to the calculations proposed in Bakshi, Kapadia, and Madan (2003). Inboth cases, the implied variances we compute are “to term,” meaning that theyare not annualized by dividing by the time remaining until expiration.

As is standard (e.g., Andersen et al. (2001)), we proxy for quadratic variationwith a realized variance measure computed as the daily sum of all squared five-minute returns along with the squared overnight return.11 Intraday returns

11 We have also examined robustness to using squared daily returns to proxy for quadraticvariation. Doing so produces equivalent results for the equity option sample but somewhat weakerresults for the S&P 500 Index option sample, though most significant coefficients remain so.


on individual equities are computed using the NYSE Trades and Quotes (TAQ)database, where we use the last recorded transaction price within each five-minute interval between 9:30 am and 4:30 pm EST, a total of 84 observationsper day for each firm.12 Realized variances for the S&P 500 Index are from theOxford-Man Realized Library, to which we add the squared overnight return.

We assess the impact of nontrading and risk premia using the followingregression:

Dt,t+1 = a + bNTt+1 + c VRPt + dNTt+1VRPt + et,t+1, (3)where Dt,t+1 now represents the average volatility differential computed fromtime t to t + 1. In this regression, NT represents a nontrading indicator, whichis equal to 1 on nontrading days and 0 otherwise, and VRP represents a vari-ance risk premium proxy similar to that proposed by Bollerslev, Tauchen, andZhou (2009). Specifically, VRP represents the demeaned difference between thesquare of the VIX Index, which is a model-free measure of implied variance,and the 22-day moving average of daily realized variances computed from five-minute index returns. Demeaning has no effect on the c or d coefficients, butit does affect the estimates of a and b in the specifications in which VRP isincluded.13

The results of these regressions are in Table V. In short, we observe a highlysignificant nontrading effect (the b coefficient) for both equity options and indexoptions. This indicates that the decrease in implied volatility remaining untilexpiration is too large over nontrading periods, implying that option pricesembed an assumption of nontrading-period variance that is significantly higherthan what is actually realized. To gauge how much higher, we can compare theestimated b coefficient, which measures the size of the nontrading effect onvariance differentials, to the average stock-level realized variance.

In our sample, the average stock-level realized variance is 0.00082, which isequivalent to an average daily volatility of 2.87%. For the same equities, esti-mates of the nontrading effect for variance differentials (b) ranges from around0.0006 to 0.0009 (note that the b coefficients in Table V have been multiplied by1,000), numbers that are approximately equal to the average realized variance.This indicates that, during the average nontrading period, the excess declinein implied variance is around one full day of realized variance. Similar resultsobtain for index options. While the b coefficient is smaller for those options, theaverage realized variance for the S&P 500 Index is only 0.00012 (an averagevolatility of 1.08%). Estimated b coefficients from index options are all larger

12 TAQ data are filtered before the five-minute returns are computed. Specifically, we excludeobservations with zero price or zero size, corrected orders, and trades with condition codes B, G, J,K, L, O, T, W, or Z. Trades on all exchanges are included. We also eliminate observations that resultin transaction-to-transaction return reversals of 25% or more and observations that are outside theCRSP daily high-low range. Finally, we compute size-weighted median prices for all transactionswith the same time stamp.

13 Demeaning the variance risk premia (VRP) is done to eliminate collinearity between thenontrading indicator and its interaction with the VRP. Without detrending, neither regressorwould be significant in the unrestricted specifications.


Table VThe Differential between Elapsed Actual and Implied Variance

This table examines the daily differentials (Dit) between elapsed realized variance (RVit) andelapsed implied variance (−�IVit). RVit is the sum of all squared returns computed over five-minute intervals within a day plus the squared overnight return. −�IVit, which measures theone-day reduction in risk-neutral variance remaining until the option’s expiration, is computedas the τ -day implied variance on day t − 1 minus the τ−1-day implied variance on day t, whereimplied variance is computed on a cumulative nonannualized basis. As described in Section II.C, weuse both the Black-Scholes model (BSIV) and the model-free approach (MFIV) of Bakshi, Kapadia,and Madan (2003) with the interpolation scheme of Hansis, Schlag, and Vilkov (2010) to computethese implied variances. In the top panel, we regress the average Dit across all stocks in the S&P500 Index on a nontrading indicator, a measure of the variance risk premium, and an interactionterm. In the lower panel, we regress the S&P 500 Index differential on the same variables. Thevariance risk premium is equal to the square of the VIX Index, converted to represent a dailyvalue, minus the 22-day moving average of realized variances on the S&P 500 Index, which arethemselves the sum of all squared five-minute returns plus the overnight return. Estimates of theintercept and the nontrading indicator have been multiplied by 1,000. Values in parentheses aret-statistics computed from Newey-West (1987) standard errors with 22 lags. Data are daily fromFebruary 2, 1996 through August 28, 2014.

Panel A: Individual Equities

Dit Based on BSIV Dit Based on MFIV

Intercept 0.1100 0.1091 0.1078 0.1520 0.1505 0.1498(2.53) (2.56) (2.58) (3.53) (3.60) (3.62)

Nontrading indicator (NT Ind) −0.9128 −0.9086 −0.9253 −0.5858 −0.5792 −0.5887(−8.76) (−8.73) (−8.39) (−7.37) (−7.28) (−7.07)

Variance risk premium (VRP) 0.8451 2.0608 1.3423 2.0399(0.84) (1.36) (1.30) (1.43)

NT Ind × VRP −5.8610 −3.3646(−1.36) (−1.16)

R2 0.0326 0.0342 0.0463 0.0166 0.0214 0.0264

Panel B: S&P 500 Index

Intercept −0.4428 −0.4400 −0.4406 −0.0611 −0.0586 −0.0590(−10.99) (−10.94) (−11.06) (−2.85) (−2.58) (−2.68)

Nontrading indicator (NT Ind) −0.2964 −0.3092 −0.3172 −0.1443 −0.1555 −0.1608(−3.46) (−3.68) (−3.64) (−2.38) (−2.63) (−2.57)

Variance risk premium (VRP) −2.7161 −2.0983 −2.4489 −2.0367(−1.78) (−1.25) (−3.75) (−2.83)

NT Ind × VRP −2.9820 −1.9906(−1.21) (−0.87)

R2 0.0033 0.0180 0.0210 0.0023 0.0387 0.0426

than this value, meaning that the decline in index implied volatilities is at leastas excessive on a relative basis. Thus, it appears as if option prices embed afull extra day of variance over each nontrading period.

Controlling for variance risk premia has little effect on the nontrading co-efficient, though in some cases, it is a highly significant predictor of variancedifferentials for the S&P 500 Index. The addition of an interaction term alsohas very little effect. Were the nontrading effect compensation for variance


risk, one might expect that it would increase when risk premia were larger.The insignificance of the interaction term therefore suggests that nontradingeffects do not stem from an aversion to variance risk. Nevertheless, we explorethis possibility more thoroughly in the following section.

III. Risk and Risk Aversion

Having established a nontrading effect in option portfolios of all types, wenow address whether the differential between trading and nontrading peri-ods is likely the result of differences in risk or risk aversion. In the results tofollow, we document that risk is indeed somewhat higher over nontrading pe-riods, which raises the possibility that the nontrading effect is attributable torisk. Unfortunately, risk adjustment for option returns is notoriously difficult,mainly due to the fact that option returns are highly non-Gaussian. We there-fore employ a number of techniques for risk adjustment that, while individuallyimperfect, together make a compelling case that risk cannot ultimately explainour findings.

We proceed by first analyzing the behavior of option trading volume and openinterest around nontrading periods, motivated by the idea that differencesin risk or risk aversion over nontrading periods should lead to predictablepatterns in trading and open interest as traders rebalance their positions. Wethen assess the risk of option returns, first by analyzing the time series, thenby examining the cross section. We find no patterns in volume or open interestthat would indicate that traders or market makers wish to reduce exposureprior to nontrading periods. In the time series, we document that, while risk ishigher over nontrading periods, various measures of risk-adjusted performanceare in agreement that the greater risk is not sufficient to justify the size ofthe nontrading returns. There is also no apparent time series variation inliquidity. Finally, in the cross section, we find that only certain types of optionrisk are rewarded in terms of more negative nontrading returns. These resultsundermine any explanation based on the Chen and Singal (2003) hypothesisthat aversion to downside risk is higher over nontrading periods. Instead,we show that the nontrading effect is consistent with the particular type ofmispricing considered by French (1984).

A. Patterns in Volume and Open Interest

If option writers are averse to holding their positions over nontrading peri-ods, then it should be reasonable to expect their positions to shrink prior tothose periods. Because options are in zero net supply, a reduction in writtenpositions means a decrease in the overall open interest in the options market.Furthermore, if option writers are particularly averse to maintaining writtenpositions in some types of contracts, say, those that are riskier in some sense,then we might expect the decline to be largest in those contracts.

To examine the behavior of open interest around nontrading periods, wecompute daily open interest, in terms of the number of option contracts, for


each of the portfolios analyzed in Table III.14 In doing so, we do not impose datafilters, such as the requirement of five consecutive days of trading activity. Wefind that open interest is not prone to measurement error, making these filtersunnecessary. More importantly, the variables on which filters are based (i.e.,volume) are too closely related to the open interest series we are tracking. Theonly filter we impose is that contracts have more than five days remaining untilexpiration. This eliminates option contracts in their final week, which wouldartificially induce a nontrading effect due to the fact that expiring contractshave zero open interest on the last day of their expiration week.

We compute two weekly open interest series for each portfolio formed bymoneyness and maturity. One is the average open interest over all days justprior to a nontrading period, which most often is simply the open interest on theFriday of that week. The other is the average open interest over all other days(usually Monday through Thursday). We then take the ratio of these two serieswithin each week and compute the time series median, which is not sensitiveto occasional outliers in the open interest ratio that are due to denominatorsnear zero.15 This median tells us how open interest rises or falls for a particularportfolio prior to nontrading periods.

The top panel of Figure 2 plots these medians, on the horizontal axis, againstthe estimated nontrading effects from Table III. If option writers are averse tokeeping positions open over nontrading periods, we would expect to see two re-sults. One is that open interest should decline prior to nontrading periods. Theother is that open interest should decline more in contracts that are particu-larly undesirable to write, which presumably are those with the most negativenontrading effects. However, the figure shows no evidence favoring either pre-diction. Open interest tends to be higher just prior to nontrading periods, byaround 2% on average. Furthermore, there is no relation between open interestand the size of the nontrading effect. Thus, if the nontrading effect is the resultof a desire to avoid risk over weekends and holidays, no attempt to avoid thatrisk by trimming positions is evident.

The bottom panel of Figure 2 plots the results of a similar examination oftrading volume. Since these ratios are all below one, it does not appear thatinvestors are motivated to reduce positions prior to nontrading periods. Fur-thermore, there is no clear relation between volume patterns and nontradingreturns.

It seems more likely that the decline in option trading is driven by the sameforces that have caused equity market volume to decline prior to weekendsand holidays for a number of decades (see Lakonishok and Maberly (1990)).Interestingly, the nontrading effect in equities documented by French (1980)has vanished in recent decades, while the volume pattern has not, makingit somewhat implausible that there is a causal relation between volume andnontrading returns in stocks either.

14 Results are robust to using dollar open interest instead.15 For most portfolios, there are no such outliers, and there is little difference between means

and medians.


Figure 2. Volume, open interest, and the nontrading effect. The plots show the relationbetween the estimated nontrading effect and volume and volatility. For each of the maturity andmoneyness-sorted portfolios analyzed in Table III, we compute the median ratio of either volumeor open interest on days prior to nontrading periods to the same variable on all other days. Theseratios appear on the horizontal axis. On the vertical axis, we show the nontrading effect estimatesfrom Table III.

B. Risk and Liquidity of Option Strategies

Table VI reports a variety of statistics describing the risk and risk-adjustedreturns corresponding to the main strategies examined thus far. One subtlechange is that we now examine excess returns from the perspective of the op-tion writer, whose returns are positive when option values decline. Flippingthe sign of the option returns is necessary for the validity of some of the perfor-mance adjustments we consider, which account for asymmetries in the returndistribution.

The first few rows of the table describe the first four moments of the threestrategies we focus on. Since option writers are short options, the nontradingeffect is shown in the table as a positive number, but the magnitudes are thesame as those reported in Table II. We now see that these higher means are as-sociated with moderately higher standard deviations, somewhat more negativeskewness, and substantially higher excess kurtosis. Few of these differencesare statistically significant, with the exception of the standard deviation and


Tab

leV

IR

isk

and

Ris

k-A

dju

sted

Per

form

ance

Th

ista

ble

repo

rts

mea

sure

sof

risk

and

retu

rnan

dof

risk

-adj

ust

edpe

rfor

man

ceof

opti

onw

riti

ng

stra

tegi

es.T

he

opti

onpo

rtfo

lios

exam

ined

are

the

sam

eas

inTa

ble

IIex

cept

that

they

are

from

the

pers

pect

ive

ofan

opti

onw

rite

rra

ther

than

buye

r.C

E(g

amm

a=

γ)r

efer

sto

the

cert

ain

ty-e

quiv

alen

tra

tefo

ra

CR

RA

inve

stor

wit

ha

risk

aver

sion

ofγ

.P

AS

and

PF

Har

eth

epe

rfor

man

cein

dice

sof

Kad

anan

dL

iu(2

014)

usi

ng

the

risk

mea

sure

sof

Au

man

nan

dS

erra

no

(200

8)an

dF

oste

ran

dH

art

(200

9),

resp

ecti

vely

.Exc

ept

for

the

mea

ns,

all

para

met

ers

are

esti

mat

edu

sin

gex

actl

yid

enti

fied

GM

Mw

ith

aN

ewey

-Wes

t(1

987)

cova

rian

cem

atri

x.t-

Sta

tist

ics

are

inpa

ren

thes

es.

Wri

tten

Un

hed

ged

Pu

tsan

dC

alls

Wri

tten

Hed

ged

Pu

tsW

ritt

enH

edge

dC

alls

Trad

ing

Non

trad

ing

Dif

fere

nce

Trad

ing

Non

trad

ing

Dif

fere

nce

Trad

ing

Non

trad

ing

Dif

fere

nce

Ave

rage

exce

ssre

turn

0.00

080.

0076

0.00

68−0

.000

30.

0082

0.00

86−0

.000

30.

0048

0.00

51(1

.76)

(9.3

0)(7

.86)

(−0.

81)

(11.

91)

(11.

93)

(−0.

69)

(5.0

8)(5

.11)

Sta

nda

rdde

viat

ion

0.02

310.

0253

0.00

220.

0196

0.02

180.

0023

0.02

180.

0322

0.01

03(3

1.49

)(1

6.00

)(1

.60)

(24.

03)

(12.

53)

(1.4

1)(1

0.49

)(6

.40)

(3.0

5)S

kew

nes

s−1

.586

5−2

.188

7−0

.602

2−2

.190

9−2

.799

8−0

.608

9−1

.233

1−2

.926

0−1

.692

9(−

9.07

)(−

4.88

)(−

1.29

)(−

6.71

)(−

3.77

)(−

0.76

)(−

2.10

)(−

1.95

)(−

0.84

)K

urt

osis

9.45

8815

.631

46.

1726

16.4

390

25.7

372

9.29

8232

.312

356

.137

723

.825

4(6

.82)

(4.7

3)(1

.72)

(4.8

1)(4

.19)

(1.3

1)(4

.34)

(4.7

8)(2

.17)

Sh

arpe

rati

o0.

0346

0.30

100.

2664

−0.0

172

0.37

630.

3935

−0.0

125

0.14

990.

1624

(1.7

0)(6

.73)

(6.0

9)(−

0.82

)(7

.05)

(7.7

7)(−

0.71

)(3

.41)

(3.6

4)A

lph

a0.

0006

0.00

760.

0070

−0.0

005

0.00

830.

0088

−0.0

008

0.00

490.

0057

(1.4

1)(1

0.01

)(8

.67)

(−1.

36)

(13.

22)

(13.

04)

(−2.

38)

(7.0

7)(7

.62)

Bet

a0.

4406

0.55

980.

1192

0.49

120.

6633

0.17

211.

1988

1.71

880.

5200

(3.5

2)(3

.93)

(0.9

2)(8

.56)

(6.0

5)(1

.65)

(11.

04)

(7.4

9)(3

.29)

Info

rmat

ion

rati

o0.

0275

0.31

730.

2898

−0.0

289

0.41

630.

4452

−0.0

471

0.22

640.

2734

(1.3

8)(7

.62)

(6.9

7)(−

1.39

)(8

.89)

(9.7

4)(−

2.57

)(4

.70)

(5.5

7)C

E(g

amm

a=

1)0.

0006

0.00

750.

0069

−0.0

005

0.00

820.

0087

−0.0

004

0.00

450.

0049

(1.3

8)(8

.90)

(8.0

2)(−

1.23

)(1

1.37

)(1

1.96

)(−

1.18

)(4

.35)

(4.2

2)C

E(g

amm

a=

3)0.

0000

0.00

680.

0068

−0.0

009

0.00

770.

0085

−0.0

010

0.00

310.

0040

(0.0

5)(7

.52)

(7.3

7)(−

2.20

)(9

.79)

(10.

86)

(−2.

30)

(2.1

3)(2

.71)

CE

(gam

ma

=10

)−0

.002

30.

0035

0.00

58−0

.002

70.

0050

0.00

77−0

.003

3−0

.014

1−0

.010

8(−

4.02

)(2

.46)

(4.1

4)(−

4.89

)(3

.67)

(5.6

1)(−

4.24

)(−

1.14

)(−

0.90

option mispricing around nontrading...

Documents